Actually, with the global convergence for convex and proximal alternating optimization given in [1]}, [2]}, our preconditioned framework can also deal with convex half quadratic models.
We next consider the convergence rate of the sequence \((u^k,y^k)\)  for our models if the KL exponent of \(L(u,y)\)  is known. This kind of convergence rate analysis is standard; see [3]}, [4]}, [5]} for more comprehensive analysis and is omitted here.
